OEIS A212558: Proof of Unproven Conjecture? Proven!
12-19-2017, 06:21 AM
Post: #9
 stored Junior Member Posts: 15 Joined: Sep 2016
RE: OEIS A212558: Proof of Unproven Conjecture? Proven!
Gerald, thanks you for the interesting quiz!

Let s(n) is the sum of the decimal digits of n.
Consider condition
4*s(n)^2 < n-s(n), (*)
4*s(n)^2 + s(n) < n

Function in left is monotonically increasing function, then
in respect that
s(n) <= 9*log10(n+1)
we get estimation
4*s(n)^2 + s(n) <= 4 * (9 * log10(n+1))^2 + 9*log10(n+1) = 324 * log10(n+1)^2 + 9*log10(n+1)

Maximal solution of the equation
324 * log10(n+1)^2 + 9*log10(n+1) = n
is n0 ~ 4313.68. (I use Wolfram Alpha for getting this value.)
Hence, condition (*) is true for all n > n0.
For n from 2999 to 4313 source statement may be checked by direct computations.
 « Next Oldest | Next Newest »

 Messages In This Thread OEIS A212558: Proof of Unproven Conjecture? Proven! - Gerald H - 12-18-2017, 08:23 AM RE: OEIS A212558: Proof of Unproven Conjecture? - pier4r - 12-18-2017, 08:26 AM RE: OEIS A212558: Proof of Unproven Conjecture? - stored - 12-18-2017, 10:21 AM RE: OEIS A212558: Proof of Unproven Conjecture? - Gerald H - 12-18-2017, 12:00 PM RE: OEIS A212558: Proof of Unproven Conjecture? - Paul Dale - 12-18-2017, 10:27 AM RE: OEIS A212558: Proof of Unproven Conjecture? - Gerald H - 12-18-2017, 01:29 PM RE: OEIS A212558: Proof of Unproven Conjecture? Proven! - Paul Dale - 12-18-2017, 10:22 PM RE: OEIS A212558: Proof of Unproven Conjecture? Proven! - Gerald H - 12-19-2017, 05:39 AM RE: OEIS A212558: Proof of Unproven Conjecture? Proven! - stored - 12-19-2017 06:21 AM RE: OEIS A212558: Proof of Unproven Conjecture? Proven! - Gerald H - 12-19-2017, 08:10 AM

User(s) browsing this thread: 1 Guest(s)